I often use the internet to find resources for learning new mathematics and due to an explosion in online activity, there is always plenty to find. Many of these turn out to be somewhat unreadable because of writing quality, organization or presentation.

I recently found out that "The Elements of Statistical Learning' by Hastie, Tibshirani and Friedman was available free online: http://www-stat.stanford.edu/~tibs/ElemStatLearn/ . It is a really well written book at a high technical level. Moreover, this is the second edition which means the book has already gone through quite a few levels of editing.

I was quite amazed to see a resource like this available free online.

Now, my question is, are there more resources like this? Are there free mathematics books that have it all: well-written, well-illustrated, properly typeset and so on?

Now, on the one hand, I have been saying 'book' but I am sure that good mathematical writing online is not limited to just books. On the other hand, I definitely don't mean the typical journal article. It's hard to come up with good criteria on this score, but I am talking about writing that is reasonably lengthy, addresses several topics and whose purpose is essentially pedagogical.

If so, I'd love to hear about them. Please suggest just one resource per comment so we can vote them up and provide a link!

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

54 Answers
54

John Baez's stuff is a fantastic resource for learning about - well, whatever John Baez is interested in, but fortunately that's a lot of interesting stuff. Scroll down for a link to TWF as well as his expository articles.

Everybody probably knows about this already, but Allen Hatcher's textbook on Algebraic Topology is excellent - clear, well-written, neatly typeset. It takes the student from basic concepts like homotopy equivalence all the way through to things like higher homotopy groups, obstruction theory and representability.

The second edition of generatingfunctionology by Herbert Wilf is freely available online and is one of my favorite math books ever. It's one of the books that made me fall in love with combinatorics (the other being the Bollobas Graph Theory book).

The preface that Donald Knuth wrote for A=B is terrific too! "Science is what we understand well enough to explain to a computer. Art is everything else we do. [\ldots] Science advances whenever an Art becomes a Science. And the state of the Art advances too, because people always leap into new territory once they have understood more about the old."
–
John SidlesMay 25 '11 at 19:20

Many know Hatchers Book, but few know the nice Concise Course in Algebraic Topology by J.P.May, which discusses, aside the standard stuff, Groupoids, Higher Homotopy and all that in a very brief and modern fashion. I think this is the book to read (for free) after/between Hatchers book.

There is also a big literature overview included, at the end of the book.

May has written much more (just look at his homepage), and I didn't read all of it. But what I read, I liked.

I was hoping that someone had posted Keith Conrad's expository stuff. Twice this week I've searched for an example in algebraic number theory (it is somewhat surprising how few of these there are in the books I own) and found the perfect answer on that page. The papers are remarkable for their high number of carefully chosen examples, just enough of which are worked out for the reader.

And just because I like the book so much, Flajolet and Sedgewick's Analytic Combinatorics is available online and is a great resource for learning about asymptotic analysis in combinatorics. The first half is also a great introduction to various techniques for writing down generating functions.

Not really pointing to a book, but I'd like to let you know I'm soon (within a month or so) launching a site dedicated to this. It is now almost finished. It is going to be a place where people can add mathematical resources, vote on them, add reviews, see other people's favorites and so on. Books will be categorized by language, level, topics, status (draft, lecture notes, books) and so on. I hope I will be able to "advertise" it trough mathoverflow: as with many "social" sites, the more people join, the more interesting it will become.

EDIT: The site is now online. It's still young, but I hope it will improve with time; I certainly have to add some features, but I decided it was time to launch and see if people actually find it useful. You can find it here

I can't believe nobody mentioned :
NUMDAM and Göttinger Digitalisierungszentrum, where you'll find digitized versions of mathematical texts... monographies and articles which made mathematical history, but sometimes still count as important references!

He has a book on buildings and many vignettes about automorphic forms, L-functions, representation theory, .... He wrote a graduate algebra book while he taught the course, and promptly got it published.

@Andrew L: "General Algebra" at the University of Minnesota covers what undergrad algebra would cover, but at a greater depth. I'd say it's sort of like the difference between "calculus" and "advanced calculus". Also, Garrett makes/grades algebra prelim exams IIRC, so the book is very good preparation. Lastly, I must say he's a very nice guy. He has a very pleasing ideology on mathematics and education (very "I want you to learn" attitude, not "I want you to get a good grade". in fact, in his $L$-functions and automorphic forms class, you get an "A", but you're still required to do work. ;)
–
QuadrescenceNov 21 '10 at 20:52

1

@AndrewL: Garrett's book, like Garrett, is unconventional. As mentioned above, the point is not to do as much algebra as is feasible in a year, the point is to do a good amount of algebra from the "right" (in the Garrett sense, whatever that means) perspective. Most mathematicians do not use category theory or homological algebra at all, and I find a first year graduate text on algebra being devoid of these topics as no great sin. Besides, only a foolish graduate student uses one algebra book.
–
Andy BNov 21 '10 at 21:13

I recommend Mel Hochster's notes. The notes for Math 614 and 615 form an introduction to commutative algebra, and 711 is on a different topic (tight closure, Henselization, etc.) every year. I think they're very easy to read.

He also has some course notes at http://www.math.umass.edu/~weston/cn.html, including truly excellent book-length notes on introductory algebraic number theory, as well as several dozen illuminating pages on local fields and ideles.